Symplectic connected-sum obstruction
= Symplectic connected-sum obstruction
A closed <symplectic manifold> of real dimension four cannot be a <connected sum> of two oriented <smooth four-manifolds> each with positive <positive index of the intersection form>. The total $b_2^+$ exceeds one; <Taubes nonvanishing theorem> and <connected-sum vanishing of the Seiberg–Witten invariant> would force the same invariant to be both nonzero and zero.
https://web.ma.utexas.edu/users/perutz/SFT6/SFT6.pdf